Thermal blow-up in a superdiffusive medium with a localized energy source is examined for a spatial domain of infinite extent in one, two, and three dimensions. An analysis of a nonlinear model of this problem reveals that the occurrence of a blow-up depends upon the spatial dimension of the superdiffusive medium. In one dimension, a blow-up always occurs. In two and three dimensions, a blow-up can be avoided if the superdiffusive properties are sufficiently enhanced. The asymptotic growth of the temperature near blow-up is determined for energy sources whose output increases in either an algebraic or an exponential manner. The results of the analysis have potential application to porous materials in which a pore filling agent is introduced to enhance the thermal transport properties.